Automatic PID Controller Tuning for Robots with Nonlinear Friction at the Joints

Transcription

1 Automatc PID Controller unng for Robots wth Nonlnear Frcton at the Jonts Abílo Azenha, Ph.D. Abstract hs paper descrbes an approach to automatc tunng of PID poston controllers for manpulators wth nonlnear frcton at the artculatons. he algorthm works n jont space coordnates and t uses the calculaton of PID controllers parameters usng step nput torue sgnals. everal smulaton results show ths scheme performance. Index erms Robot manpulators; tunng; frcton; tme doman analyss; smulaton. I. INRODUCION In ths artcle an algorthm for the trackng control of robot manpulators s tested. he motvaton for ths work s the need for accurate poston control n tasks such as assembly, deburrng, pantng, weldng, etc. just before the robot end-effector contact wth the envronment. On the other hand, the frcton phenomenon s always present n such systems, at least n the arm jonts. hs leads to detrmental aspects n the systems performance [1-7], because for small veloctes the stck-slp effect arses. In ths lne of thought, t s worthwhle to take the nonlnear frcton phenomenon nto consderaton. Another ueston s how to choose the sutable system controller n terms of smplcty, practcablty and performance. Of course, the classcal PID controller s the frst choce, because of ts smplcty and practcablty. Nevertheless, t remans the problem of the parameters tunng, beng the desrable soluton an automatc scheme (.e. wthout the user s need for extra calculatons). In ths paper the open-loop Zegler-Nchols [8] approach s used, whch was orgnally employed n the rough estmaton of PID controllers parameters for smple sngle-nput-sngle-output (IO) lumpedcharacterstcs systems. ome other authors desgned PID controllers recently (e.g. [9-11]) wth dfferent objectves and resources n mnd, beng the systems performance a common fnal effort. In ths lne of thought, the R robot wth nonlnear frcton at the jonts and n poston control (.e. wthout constrant Department of Electrcal and Computer Engneerng, Faculty of Engneerng, Unversty of Porto, Rua Dr. Roberto Fras, Porto, Portugal, Phone: (ext.131), Fax: , E-mal: azenha@fe.up.pt surfaces) s employed as prototype manpulator. he frcton model comprses stcton, Coulomb, vscous and trbeck effects to smulate effcently the low velocty behavor. he arm s moved to the neghborhood of the task operatng pont and then two nput steps are appled (one at a tme and n the respectve jont torue) for the automatc PID parameters tunng. Next the PID controllers parameters are calculated numercally by the (dgtal) control system usng the classcal Zegler-Nchols procedure. he multple-nput-multple-output (MIMO) system characterstc and the manpulator gravtatonal dynamc terms are taken nto consderaton. he artcle s organzed as follows. ecton two descrbes the models employed n ths work. ecton three presents the auto-tunng algorthm and secton four shows some smulaton results. Fnally, secton fve outlnes the man conclusons. II. YEM MODELING In ths secton the dynamc models for both the deal robot and the nonlnear frcton to be accounted for n the arm jont torues are presented. A. Ideal rgd-lnk-rgd-jont robot he dynamc euaton of an deal rgd-lnk-rgd-jont robot wth n lnks s: τ = H ( && c(, & ) g( (1) Here τ s the n 1 vector of actuator torues, s the n 1 vector of jont coordnates, H( s the n n nerta matrx, c (, &) s the n 1 vector of centrfugal/corols terms and g( s the n 1 vector of gravtatonal effects. In ths study the R robot wll be adopted as prototype manpulator, wth dynamcs gven by: ( ) m 1 m r1 mr mr mrrc 1 H ( ) = mrrc J J (a) 1 1m 1g mr mrrc mr J J 1 m g & & & c,& ( ) = mrr 1 mrr 1 1 (b) & mrr 1 1 page 1 of 6

2 ( ) gmrc mrc mrc g ( ) = gmrc1 where C cos( ) = sn =, Cj = ( j ) ( ) j j (c) cos, ( ) = sn,. he manpulator s depcted n Fg. 1 whle the numercal values adopted for the R robot are presented n able I. B. Nonlnear Frcton he frcton model, used n all the smulatons, present n the contact of a mass M wth other mass and wth relatve velocty x s depcted n Fg.. It comprses four components: ) stcton (statc frcton); ) Coulomb frcton; ) vscous frcton; v) trbeck effect. he stcton represents the phenomenon near the orgn and t s smulated through the Karnopp s [1] algorthm (Fg. 3). he Coulomb part s related to a constant frcton force (K C ) as a functon of relatve velocty ( x = V ). he vscous component s a force that s proportonal to the relatve velocty (wth proportonalty constant B). Fnally, the trbeck effect stands for an exponental behavor near the orgn and can be defned by the followng euaton (3): F trbeck x x e = F (3) where x s the trbeck velocty constant. All the prevous components are summed up for the frcton model beng sutable for the study and the mathematcal model s correspondent to the graph of Fg.. III. CONROLLER UNING In ths secton the Zegler-Nchols method s shown how to be used to tune a PID controller, and how ths method wll be appled to robot manpulators s explaned. tablty consderatons are also made. A. Zegler-Nchols Open-Loop Method he Zegler-Nchols open-loop method [8] assumes that the process to be controlled has the followng transfer functon (4): K pe s s (4) where K p s calculated as the maxmum value of the unt step system response slope and s the correspondng tme delay. hen, the PID parameters are obtaned through the set of formulae (5): 1. K = (5a) R r = (5b) d =. 5 (5c) where R r s called the reacton rate and s assumed to be eual to K p. he tme response of the employed PID controller s (6): m dc () t K e e() τ dτ 1 = (6) d dt where c(t) s the process output, e(t) s the error sgnal and m(t) s the controller output. B. Applcaton to Robot Manpulators Poston Control In ths study, the parameters and R r are found numercally, through applcaton of a step test nput (wth duraton t t =.5 s) when the system s at rest and n open-loop mode. hs test nput s calculated by euaton (7): τ = g ( ) δjh j ( t), = 1,..., n, j = 1,..., n (7) where = [ 1,,..., n ] s the ntal jont operatng pont and h j (t) s the nput step appled at t = s. Note that the magntude of h j (t) must be greater then the stcton torue. he δ j coeffcent s defned by (8): δ j 1, =, = j j (8) he j ndex s the test number (j = 1,...n),.e. the total number of tests s eual to the jonts number n. In ths lne of thought, R rj s the maxmum dervatve of the output x j (t) = j (t) j and dvded by the ampltude of the step nput (f dfferent than unty). Fnally, the parameter j s obtaned by the followng expresson (9), where all the rght-hand values are obtaned n the pont correspondng to the greatest system output slope: x j1 j = t j1 (9) x j1 hs last euaton can be llustrated by the plot n Fg. 4. Usng ths scheme, the decentralzed PID controllers parameters K j, j and dj (j = 1,...,n) are computed wth ther respectve dentfed values of j and R rj (j = 1,...,n). page of 6

3 C. tablty Analyss ome stablty ssues wll be carred out through the Lyapunov method. Consder the set of dynamc euatons descrbng ths system: τ = H( c(, g( τ τ = K frcton [ dτ d ] 1 ( r ( r ( τ) (,, H( > (1) where K, and d are the n n dagonal PID parameters matrces, and r s the jont reference nput vector. Let be V a Lyapunov canddate functon: where: 1 (11) V = H() P() (), g() = P (1a) (1b) P = g () () he tme dervatve of V becomes [13], usng expressons (1) and (1): d 1 V = H dt () g()= = [ τ τ (, )]= frcton t 1 ( ( ( φ) dφ = K r KΤ τ [ (, )] d r frcton (13) Analyzng euaton (13) we conclude that for better system stablty, the PID controller parameters must be such that K jj s small, beng jj and djj (j = 1,...,n) large. In the next secton these arguments are confrmed. IV. REUL In ths secton two smulaton examples for the R robot wth nonlnear frcton at the jonts are presented. he numercal values tested are (wth varables translated from lnear to rotatonal moton): = 15π/36 rad, K C = 5 Nm, B =.5 Nm/s, Dω =.1 rad/s, τ H = 6 Nm, =. rad/s, τ = 1 Nm, h (t) = 1 Nm, = 1,. he tested system s presented n Fg. 5 n block dagram form. In ths lne of thought, Fg. 6 shows a smulaton where the nputs of the MIMO unty feedback system are r1 = 1.1 rad and r =. he respectve auto-tuned PID controllers parameters are K 1 = 719 Nm, d1 = 1 =.59 s, K = 778 Nm, d = =.6 s. Note that the eualty d = ( = 1,...,n) s always employed, because the smulatons results showed always the better system performance under ths condton. hs agrees wth last secton stablty results, because d s rased some order of magntude [see euatons (5b-c)]. Fg. 7 depcts the second smulaton here for r1 = 1 and r =.1 rad. Obvously, the PID controllers parameters are the same as for the frst smulaton, because the physcal system s the same. It s observed that the steady-state error sgnal s very small ( has an error even lower than 1 ) and the overshoot s not sgnfcant. he sgnals n the loops are all somehow smooth. V. CONCLUION hs paper studed the auto-tunng of PID controllers for robot manpulators subjected to nonlnear frcton at the jonts. he frcton model used the more mportant aspects for a suffcently detaled compromse between realty and smplcty. he obtaned results are good, for an accurate poston control wthout the need for tral-and-error PID parameters tunng, as ntally planned. ACKNOWLEDGMEN he author s thankful to Prof. Adrano Carvalho for hs encouragement to the wrtng of ths paper. REFERENCE [1] B. Armstrong-Hélouvry, P. Dupont and C. Canudas-de-Wt, A urvey of Models, Analyss ools and Compensaton Methods for the Control of Machnes wth Frcton, Automatca, vol. 3, no. 7, pp , [] A. Azenha and J.A.. Machado, Varable tructure Control of Robots wth Nonlnear Frcton and Backlash at the Jonts, n Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton, Mnneapols, UA, 1996, pp [3] A. Azenha and J.A.. Machado, On the Descrbng Functon Method and the Predcton of Lmt Cycles n Nonlnear Dynamcal ystems, Journal of ystems Analyss, Modelng and mulaton, vol. 33, no. 3, pp. 37-3, [4] M.R. Popovc and A.A. Goldenberg, Modelng of Frcton Usng pectral Analyss, IEEE ransactons on Robotcs and Automaton, vol. 14, no. 1, pp , [5] H. Du and.. Nar, Modelng and Compensaton of Low-Velocty Frcton wth Bounds, IEEE ransactons on Control page 3 of 6

4 ystems echnology, vol. 7, no. 1, pp , [6] P. Lschnsky, C. Canudas-de-Wt and G. Morel, Frcton Compensaton for an Industral Hydraulc Robot, IEEE Control ystems, vol. 19, no. 1, pp. 5-3, [7] P. ome, Robust Adaptve Frcton Compensaton for rackng Control of Robot Manpulators, IEEE ransactons on Automatc Control, vol. 45, no. 11, pp ,. [8] J.L.M. de Carvalho, Dynamcal ystems and Automatc Control, Prentce-Hall Internatonal, London, [9] M. Kawafuku, M. asak and. Kato, elf- unng PID Control of a Flexble Mcro- Actuator usng Neural Networks, n Proceedngs of the IEEE Internatonal Conference on ystems, Man, and Cybernetcs, La Jolla, an Dego, Calforna, UA, 1998, pp [1] Q.-G. Wang,.-H. Lee, H.-W. Fung and Y. Zhang, PID unng for Improved Performance, IEEE ransactons on Control ystems echnology, vol. 7, no. 4, pp , [11] B. Armstrong, D. Neevel and. Kusk, New Results n NPID Control: rackng, Integral Control, Frcton Compensaton and Expermental Results, IEEE ransactons on Control ystems echnology, vol. 9, no., pp , 1. [1] D. Karnopp, Computer mulaton of tck- lp Frcton n Mechancal Dynamc ystems, AME Journal of Dynamc ystems, Measurement, and Control, vol. 17, no. 1, pp. 1-13, [13] R. Lozano, A. Valera, P. Albertos,. Armoto and. Nakayama, PD Control of Robot Manpulators wth Jont Flexblty, Actuators Dynamcs and Frcton, Automatca, vol. 35, no. 1, pp , ABLE I: he R robot parameters. Lnk m (Kg) r (m) J m (Kgm ) J g (Kgm ) y m r J 1g r 1 J g 1 m 1 F frcton K C B J 1m Fg. 1: he R robot. J m x x K C B DV Fg.. Plot of the frcton model. page 4 of 6

5 P V dt DP DP P P F V DP = M DV F frcton DV 1 DV DV DV F slpng F f F H F stck F H Fg. 3. Block dagram of the Karnopp s stck-slp moton smulaton algorthm. xj(t) x j1 t j1 t Fg. 4. Graph for determnng parameter j (j = 1,...,n). r Error gnal Controller Output Process Output e Decentralzed PID Controller τ Robot wth Nonlnear Jont Frcton Fg. 5. Feedback control system. page 5 of 6

6 e1 (rad/s) e1 (rad) e (rad/s) e (rad) a) τ1 (Nm ) tm e (s ) τ (Nm ) tm e (s ) b) Fg. 6. me responses for the R robot under poston control and the auto-tunng PID scheme (K 1 = 719 Nm, d1 = 1 =.59 s, K = 778 Nm, d = =.6 s, zero ncremental nput δ r = r ): a) Phase-plane, b) orue nputs. e1 (rad/s).1 e (rad/s) e1 (rad) e (rad) τ1 (Nm) tme (s) a) τ (Nm) tme (s) b) Fg. 7. me responses for the R robot under poston control and the auto-tunng PID scheme (K 1 = 719 Nm, d1 = 1 =.59 s, K = 778 Nm, d = =.6 s, zero ncremental nput δ r1 = r1 1 ): a) Phase-plane, b) orue nputs. page 6 of 6